{"id":13679,"date":"2018-08-23T21:37:09","date_gmt":"2018-08-23T21:37:09","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/precalcone\/?post_type=chapter&#038;p=13679"},"modified":"2025-02-05T05:18:02","modified_gmt":"2025-02-05T05:18:02","slug":"rates-of-change-and-behavior-of-graphs","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/precalculus\/chapter\/rates-of-change-and-behavior-of-graphs\/","title":{"raw":"Rates of Change and Behavior of Graphs","rendered":"Rates of Change and Behavior of Graphs"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Find the average rate of change of a function.<\/li>\r\n \t<li>Use a graph to determine where a function is increasing, decreasing, or constant.<\/li>\r\n \t<li>Use a graph to locate local and absolute maxima and local minima.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<p id=\"fs-id1165135194500\">Gasoline costs have experienced some wild fluctuations over the last several decades. The table below[footnote]http:\/\/www.eia.gov\/totalenergy\/data\/annual\/showtext.cfm?t=ptb0524. Accessed 3\/5\/2014.[\/footnote]\u00a0lists the average cost, in dollars, of a gallon of gasoline for the years 2005\u20132012. The cost of gasoline can be considered as a function of year.<\/p>\r\n\r\n<table summary=\"Two rows and nine columns. The first row is labeled, \"><colgroup> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/><\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td><strong>[latex]y[\/latex]<\/strong><\/td>\r\n<td>2005<\/td>\r\n<td>2006<\/td>\r\n<td>2007<\/td>\r\n<td>2008<\/td>\r\n<td>2009<\/td>\r\n<td>2010<\/td>\r\n<td>2011<\/td>\r\n<td>2012<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>[latex]C\\left(y\\right)[\/latex]<\/strong><\/td>\r\n<td>2.31<\/td>\r\n<td>2.62<\/td>\r\n<td>2.84<\/td>\r\n<td>3.30<\/td>\r\n<td>2.41<\/td>\r\n<td>2.84<\/td>\r\n<td>3.58<\/td>\r\n<td>3.68<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1165133097252\">If we were interested only in how the gasoline prices changed between 2005 and 2012, we could compute that the cost per gallon had increased from $2.31 to $3.68, an increase of $1.37. While this is interesting, it might be more useful to look at how much the price changed <em>per year<\/em>. In this section, we will investigate changes such as these.<\/p>\r\n\r\n<h2>Finding the Average Rate of Change of a Function<\/h2>\r\n<p id=\"fs-id1165137834011\">The price change per year is a <strong>rate of change<\/strong> because it describes how an output quantity changes relative to the change in the input quantity. We can see that the price of gasoline in\u00a0the table above\u00a0did not change by the same amount each year, so the rate of change was not constant. If we use only the beginning and ending data, we would be finding the <strong>average rate of change<\/strong> over the specified period of time. To find the average rate of change, we divide the change in the output value by the change in the input value.<\/p>\r\n\r\n<div id=\"fs-id1165135452482\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{align}\\text{Average rate of change}&amp;=\\frac{\\text{Change in output}}{\\text{Change in input}} \\\\[1mm] &amp;=\\frac{\\Delta y}{\\Delta x} \\\\[1mm] &amp;= \\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}} \\\\[1mm] &amp;= \\frac{f\\left({x}_{2}\\right)-f\\left({x}_{1}\\right)}{{x}_{2}-{x}_{1}}\\end{align}[\/latex]<\/div>\r\n<p id=\"fs-id1165135471272\">The Greek letter [latex]\\Delta [\/latex] (delta) signifies the change in a quantity; we read the ratio as \"delta-<em>y<\/em> over delta-<em>x<\/em>\" or \"the change in [latex]y[\/latex] divided by the change in [latex]x[\/latex].\" Occasionally we write [latex]\\Delta f[\/latex] instead of [latex]\\Delta y[\/latex], which still represents the change in the function\u2019s output value resulting from a change to its input value. It does not mean we are changing the function into some other function.<\/p>\r\n<p id=\"fs-id1165137539940\">In our example, the gasoline price increased by $1.37 from 2005 to 2012. Over 7 years, the average rate of change was<\/p>\r\n\r\n<div id=\"fs-id1165137526960\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\frac{\\Delta y}{\\Delta x}=\\frac{{1.37}}{\\text{7 years}}\\approx 0.196\\text{ dollars per year}[\/latex]<\/div>\r\n<p id=\"fs-id1165137418924\">On average, the price of gas increased by about 19.6\u00a2 each year.<\/p>\r\n<p id=\"fs-id1165135397217\">Other examples of rates of change include:<\/p>\r\n\r\n<ul id=\"fs-id1165137424067\">\r\n \t<li>A population of rats increasing by 40 rats per week<\/li>\r\n \t<li>A car traveling 68 miles per hour (distance traveled changes by 68 miles each hour as time passes)<\/li>\r\n \t<li>A car driving 27 miles per gallon (distance traveled changes by 27 miles for each gallon)<\/li>\r\n \t<li>The current through an electrical circuit increasing by 0.125 amperes for every volt of increased voltage<\/li>\r\n \t<li>The amount of money in a college account decreasing by $4,000 per quarter<\/li>\r\n<\/ul>\r\n<div class=\"textbox\">\r\n<h3 class=\"title\">A General Note: Rate of Change<\/h3>\r\n<p id=\"fs-id1165137780744\">A rate of change describes how an output quantity changes relative to the change in the input quantity. The units on a rate of change are \"output units per input units.\"<\/p>\r\n<p id=\"fs-id1165137544638\">The average rate of change between two input values is the total change of the function values (output values) divided by the change in the input values.<\/p>\r\n\r\n<div id=\"fs-id1165134060431\" class=\"equation\" style=\"text-align: center;\">[latex]\\frac{\\Delta y}{\\Delta x}=\\frac{f\\left({x}_{2}\\right)-f\\left({x}_{1}\\right)}{{x}_{2}-{x}_{1}}[\/latex]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135530407\" class=\"note precalculus howto textbox\">\r\n<h3 id=\"fs-id1165137762240\">How To: Given the value of a function at different points, calculate the average rate of change of a function for the interval between two values [latex]{x}_{1}[\/latex] and [latex]{x}_{2}[\/latex].<\/h3>\r\n<ol id=\"fs-id1165137442714\">\r\n \t<li>Calculate the difference [latex]{y}_{2}-{y}_{1}=\\Delta y[\/latex].<\/li>\r\n \t<li>Calculate the difference [latex]{x}_{2}-{x}_{1}=\\Delta x[\/latex].<\/li>\r\n \t<li>Find the ratio [latex]\\frac{\\Delta y}{\\Delta x}[\/latex].<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_01_03_01\" class=\"example\">\r\n<div id=\"fs-id1165135485962\" class=\"exercise\">\r\n<div id=\"fs-id1165137464225\" class=\"problem textbox shaded\">\r\n<h3>Example 1: Computing an Average Rate of Change<\/h3>\r\n<p id=\"fs-id1165137603118\">Using the data in the table below, find the average rate of change of the price of gasoline between 2007 and 2009.<\/p>\r\n\r\n<table summary=\"Two rows and nine columns. The first row is labeled, \"><colgroup> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/><\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td><strong>[latex]y[\/latex]<\/strong><\/td>\r\n<td>2005<\/td>\r\n<td>2006<\/td>\r\n<td>2007<\/td>\r\n<td>2008<\/td>\r\n<td>2009<\/td>\r\n<td>2010<\/td>\r\n<td>2011<\/td>\r\n<td>2012<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>[latex]C\\left(y\\right)[\/latex]<\/strong><\/td>\r\n<td>2.31<\/td>\r\n<td>2.62<\/td>\r\n<td>2.84<\/td>\r\n<td>3.30<\/td>\r\n<td>2.41<\/td>\r\n<td>2.84<\/td>\r\n<td>3.58<\/td>\r\n<td>3.68<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"562005\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"562005\"]\r\n<p id=\"fs-id1165135209401\">In 2007, the price of gasoline was $2.84. In 2009, the cost was $2.41. The average rate of change is<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{align}\\frac{\\Delta y}{\\Delta x}&amp;=\\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}} \\\\[1mm] &amp;=\\frac{2.41-2.84}{2009 - 2007} \\\\[1mm] &amp;=\\frac{-0.43}{2\\text{ years}} \\\\[1mm] &amp;={-0.22}\\text{ per year}\\end{align}[\/latex]<\/p>\r\n\r\n<h4>Analysis of the Solution<\/h4>\r\n<p id=\"fs-id1165137784092\">Note that a decrease is expressed by a negative change or \"negative increase.\" A rate of change is negative when the output decreases as the input increases or when the output increases as the input decreases.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe following video provides another example of how to find the average rate of change between two points from a table of values.\r\n\r\nhttps:\/\/www.youtube.com\/watch?v=iJ_0nPUUlOg&amp;feature=youtu.be\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1165135160759\">Using the data in the table below,\u00a0find the average rate of change between 2005 and 2010.<\/p>\r\n\r\n<table summary=\"Two rows and nine columns. The first row is labeled, \">\r\n<tbody>\r\n<tr>\r\n<td><strong>[latex]y[\/latex]<\/strong><\/td>\r\n<td>2005<\/td>\r\n<td>2006<\/td>\r\n<td>2007<\/td>\r\n<td>2008<\/td>\r\n<td>2009<\/td>\r\n<td>2010<\/td>\r\n<td>2011<\/td>\r\n<td>2012<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>[latex]C\\left(y\\right)[\/latex]<\/strong><\/td>\r\n<td>2.31<\/td>\r\n<td>2.62<\/td>\r\n<td>2.84<\/td>\r\n<td>3.30<\/td>\r\n<td>2.41<\/td>\r\n<td>2.84<\/td>\r\n<td>3.58<\/td>\r\n<td>3.68<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"175600\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"175600\"]\r\n\r\n[latex]\\dfrac{$2.84-$2.31}{5\\text{ years}}=\\dfrac{$0.53}{5\\text{ years}}=$0.106[\/latex] per year.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div id=\"Example_01_03_02\" class=\"example\">\r\n<div id=\"fs-id1165137851963\" class=\"exercise\">\r\n<div id=\"fs-id1165137437853\" class=\"problem textbox shaded\">\r\n<h3>Example 2: Computing Average Rate of Change from a Graph<\/h3>\r\nGiven the function [latex]g\\left(t\\right)[\/latex] shown in Figure 1, find the average rate of change on the interval [latex]\\left[-1,2\\right][\/latex].\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010553\/CNX_Precalc_Figure_01_03_0012.jpg\" alt=\"Graph of a parabola.\" width=\"487\" height=\"295\" \/> <b>Figure 1<\/b>[\/caption]\r\n\r\n[reveal-answer q=\"806109\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"806109\"]\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010553\/CNX_Precalc_Figure_01_03_0022.jpg\" alt=\"Graph of a parabola with a line from points (-1, 4) and (2, 1) to show the changes for g(t) and t.\" width=\"487\" height=\"296\" \/> <b>Figure 2<\/b>[\/caption]\r\n\r\nAt [latex]t=-1[\/latex], the graph\u00a0shows [latex]g\\left(-1\\right)=4[\/latex]. At [latex]t=2[\/latex], the graph shows [latex]g\\left(2\\right)=1[\/latex].<span id=\"fs-id1165137387448\">\r\n<\/span>\r\n<p id=\"fs-id1165137591169\">The horizontal change [latex]\\Delta t=3[\/latex] is shown by the red arrow, and the vertical change [latex]\\Delta g\\left(t\\right)=-3[\/latex] is shown by the turquoise arrow. The output changes by \u20133 while the input changes by 3, giving an average rate of change of<\/p>\r\n<p style=\"text-align: center;\">[latex]\\frac{1 - 4}{2-\\left(-1\\right)}=\\frac{-3}{3}=-1[\/latex]<\/p>\r\n\r\n<h4>Analysis of the Solution<\/h4>\r\n<p id=\"fs-id1165135538482\">Note that the order we choose is very important. If, for example, we use [latex]\\frac{{y}_{2}-{y}_{1}}{{x}_{1}-{x}_{2}}[\/latex], we will not get the correct answer. Decide which point will be 1 and which point will be 2, and keep the coordinates fixed as [latex]\\left({x}_{1},{y}_{1}\\right)[\/latex] and [latex]\\left({x}_{2},{y}_{2}\\right)[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"Example_01_03_03\" class=\"example\">\r\n<div id=\"fs-id1165135536188\" class=\"exercise\">\r\n<div id=\"fs-id1165137835656\" class=\"problem textbox shaded\">\r\n<h3>Example 3: Computing Average Rate of Change from a Table<\/h3>\r\n<p id=\"fs-id1165135515898\">After picking up a friend who lives 10 miles away, Anna records her distance from home over time. The values are shown in the table below.\u00a0Find her average speed over the first 6 hours.<\/p>\r\n\r\n<table id=\"Table_01_03_02\" summary=\"Two rows and nine columns. The first row is labeled, \"><colgroup> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/><\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td><strong><em>t<\/em> (hours)<\/strong><\/td>\r\n<td>0<\/td>\r\n<td>1<\/td>\r\n<td>2<\/td>\r\n<td>3<\/td>\r\n<td>4<\/td>\r\n<td>5<\/td>\r\n<td>6<\/td>\r\n<td>7<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong><em>D<\/em>(<em>t<\/em>) (miles)<\/strong><\/td>\r\n<td>10<\/td>\r\n<td>55<\/td>\r\n<td>90<\/td>\r\n<td>153<\/td>\r\n<td>214<\/td>\r\n<td>240<\/td>\r\n<td>292<\/td>\r\n<td>300<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"566859\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"566859\"]\r\n<p id=\"fs-id1165137891478\">Here, the average speed is the average rate of change. She traveled 282 miles in 6 hours, for an average speed of<\/p>\r\n<p style=\"text-align: center;\">[latex]\\frac{292 - 10}{6 - 0} =\\frac{282}{6} =47[\/latex]<\/p>\r\n<p id=\"fs-id1165135400200\">The average speed is 47 miles per hour.<\/p>\r\n\r\n<h4>Analysis of the Solution<\/h4>\r\n<p id=\"fs-id1165137731074\">Because the speed is not constant, the average speed depends on the interval chosen. For the interval [2,3], the average speed is 63 miles per hour.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"Example_01_03_04\" class=\"example\">\r\n<div id=\"fs-id1165135353057\" class=\"exercise\">\r\n<div id=\"fs-id1165135383644\" class=\"problem textbox shaded\">\r\n<h3>Example 4: Computing Average Rate of Change for a Function Expressed as a Formula<\/h3>\r\n<p id=\"fs-id1165131958324\">Compute the average rate of change of [latex]f\\left(x\\right)={x}^{2}-\\frac{1}{x}[\/latex] on the interval [latex]\\text{[2,}\\text{4].}[\/latex]<\/p>\r\n[reveal-answer q=\"222718\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"222718\"]\r\n<p id=\"fs-id1165137595441\">We can start by computing the function values at each <strong>endpoint<\/strong> of the interval.<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{align}f\\left(2\\right)&amp;={2}^{2}-\\frac{1}{2} &amp;&amp;&amp; f\\left(4\\right)&amp;={4}^{2}-\\frac{1}{4} \\\\ &amp;=4-\\frac{1}{2} &amp;&amp;&amp;&amp; =16-{1}{4} \\\\ &amp;=\\frac{7}{2} &amp;&amp;&amp;&amp; =\\frac{63}{4} \\end{align}[\/latex]<\/p>\r\n<p id=\"fs-id1165137427523\">Now we compute the average rate of change.<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{align}\\text{Average rate of change}&amp;=\\frac{f\\left(4\\right)-f\\left(2\\right)}{4 - 2} \\\\[1mm] &amp;=\\frac{\\frac{63}{4}-\\frac{7}{2}}{4 - 2} \\\\[1mm] &amp;=\\frac{\\frac{49}{4}}{2} \\\\[1mm] &amp;=\\frac{49}{8} \\end{align}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\nThe following video provides another example of finding the average rate of change of a function given a formula and an interval.\r\n\r\nhttps:\/\/www.youtube.com\/watch?v=g93QEKJXeu4&amp;feature=youtu.be\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1165137832324\">Find the average rate of change of [latex]f\\left(x\\right)=x - 2\\sqrt{x}[\/latex] on the interval [latex]\\left[1,9\\right][\/latex].<\/p>\r\n[reveal-answer q=\"191250\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"191250\"]\r\n\r\n[latex]\\frac{1}{2}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\n[ohm_question hide_question_numbers=1]165703[\/ohm_question]\r\n\r\n<\/div>\r\n<div id=\"Example_01_03_05\" class=\"example\">\r\n<div id=\"fs-id1165137772170\" class=\"exercise\">\r\n<div id=\"fs-id1165137772173\" class=\"problem textbox shaded\">\r\n<h3>Example 5: Finding the Average Rate of Change of a Force<\/h3>\r\n<p id=\"fs-id1165135443718\">The <strong>electrostatic force<\/strong> [latex]F[\/latex], measured in newtons, between two charged particles can be related to the distance between the particles [latex]d[\/latex], in centimeters, by the formula [latex]F\\left(d\\right)=\\frac{2}{{d}^{2}}[\/latex]. Find the average rate of change of force if the distance between the particles is increased from 2 cm to 6 cm.<\/p>\r\n[reveal-answer q=\"271117\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"271117\"]\r\n<p id=\"fs-id1165137770364\">We are computing the average rate of change of [latex]F\\left(d\\right)=\\frac{2}{{d}^{2}}[\/latex] on the interval [latex]\\left[2,6\\right][\/latex].<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{align}\\text{Average rate of change }&amp;=\\frac{F\\left(6\\right)-F\\left(2\\right)}{6 - 2} \\\\[1mm] &amp;=\\frac{\\frac{2}{{6}^{2}}-\\frac{2}{{2}^{2}}}{6 - 2} &amp;&amp; \\text{Simplify}. \\\\[1mm] &amp;=\\frac{\\frac{2}{36}-\\frac{2}{4}}{4} \\\\[1mm] &amp;=\\frac{-\\frac{16}{36}}{4} &amp;&amp;\\text{Combine numerator terms}. \\\\[1mm] &amp;=-\\frac{1}{9}&amp;&amp;\\text{Simplify}\\end{align}[\/latex]<\/p>\r\n<p id=\"fs-id1165135543242\">The average rate of change is [latex]-\\frac{1}{9}[\/latex] newton per centimeter.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"Example_01_03_06\" class=\"example\">\r\n<div id=\"fs-id1165135174952\" class=\"exercise\">\r\n<div id=\"fs-id1165135174954\" class=\"problem textbox shaded\">\r\n<h3>Example 6: Finding an Average Rate of Change as an Expression<\/h3>\r\n<p id=\"fs-id1165135155397\">Find the average rate of change of [latex]g\\left(t\\right)={t}^{2}+3t+1[\/latex] on the interval [latex]\\left[0,a\\right][\/latex]. The answer will be an expression involving [latex]a[\/latex].<\/p>\r\n[reveal-answer q=\"138559\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"138559\"]\r\n<p id=\"fs-id1165137418913\">We use the average rate of change formula.<\/p>\r\n<p style=\"text-align: center;\">\u200b[latex]\\begin{align}\\text{Average rate of change}&amp;=\\frac{g\\left(a\\right)-g\\left(0\\right)}{a - 0}&amp;&amp;\\text{Evaluate} \\\\[1mm] &amp;\u200b=\\frac{\\left({a}^{2}+3a+1\\right)-\\left({0}^{2}+3\\left(0\\right)+1\\right)}{a - 0}&amp;&amp;\\text{Simplify}\u200b \\\\[1mm] &amp;=\\frac{{a}^{2}+3a+1 - 1}{a}&amp;&amp;\\text{Simplify and factor}\u200b \\\\[1mm] &amp;=\\frac{a\\left(a+3\\right)}{a}&amp;&amp;\\text{Divide by the common factor }a\u200b \\\\[1mm] &amp;=a+3 \\end{align}[\/latex]<\/p>\r\n<p id=\"fs-id1165133316469\">This result tells us the average rate of change in terms of [latex]a[\/latex] between [latex]t=0[\/latex] and any other point [latex]t=a[\/latex]. For example, on the interval [latex]\\left[0,5\\right][\/latex], the average rate of change would be [latex]5+3=8[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1165134149846\">Find the average rate of change of [latex]f\\left(x\\right)={x}^{2}+2x - 8[\/latex] on the interval [latex]\\left[5,a\\right][\/latex].<\/p>\r\n[reveal-answer q=\"944512\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"944512\"]\r\n\r\n[latex]a+7[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<p id=\"fs-id1165137784644\">As part of exploring how functions change, we can identify intervals over which the function is changing in specific ways. We say that a function is increasing on an interval if the function values increase as the input values increase within that interval. Similarly, a function is decreasing on an interval if the function values decrease as the input values increase over that interval. The average rate of change of an increasing function is positive, and the average rate of change of a decreasing function is negative. Figure 3\u00a0shows examples of increasing and decreasing intervals on a function.<\/p>\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010554\/CNX_Precalc_Figure_01_03_0042.jpg\" alt=\"Graph of a polynomial that shows the increasing and decreasing intervals and local maximum and minimum.\" width=\"487\" height=\"518\" \/>\r\n<p style=\"text-align: center;\"><strong>Figure 3.<\/strong> The function [latex]f\\left(x\\right)={x}^{3}-12x[\/latex] is increasing on [latex]\\left(-\\infty \\text{,}-\\text{2}\\right){{\\cup }^{\\text{ }}}^{\\text{ }}\\left(2,\\infty \\right)[\/latex] and is decreasing on [latex]\\left(-2\\text{,}2\\right)[\/latex].<\/p>\r\nThis video further explains how to find where a function is increasing or decreasing.\r\n\r\nhttps:\/\/www.youtube.com\/watch?v=78b4HOMVcKM\r\n<p id=\"fs-id1165134272749\">While some functions are increasing (or decreasing) over their entire domain, many others are not. A value of the input where a function changes from increasing to decreasing (as we go from left to right, that is, as the input variable increases) is called a <strong>local maximum<\/strong>. If a function has more than one, we say it has local maxima. Similarly, a value of the input where a function changes from decreasing to increasing as the input variable increases is called a <strong>local minimum<\/strong>. The plural form is \"local minima.\" Together, local maxima and minima are called <strong>local extrema<\/strong>, or local extreme values, of the function. (The singular form is \"extremum.\") Often, the term <em>local<\/em> is replaced by the term <em>relative<\/em>. In this text, we will use the term <em>local<\/em>.<\/p>\r\n<p id=\"fs-id1165134547216\">Clearly, a function is neither increasing nor decreasing on an interval where it is constant. A function is also neither increasing nor decreasing at extrema. Note that we have to speak of <em>local<\/em> extrema, because any given local extremum as defined here is not necessarily the highest maximum or lowest minimum in the function\u2019s entire domain.<\/p>\r\nFor the function in Figure 4, the local maximum is 16, and it occurs at [latex]x=-2[\/latex]. The local minimum is [latex]-16[\/latex] and it occurs at [latex]x=2[\/latex].\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"731\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010554\/CNX_Precalc_Figure_01_03_0142.jpg\" alt=\"Graph of a polynomial that shows the increasing and decreasing intervals and local maximum and minimum. The local maximum is 16 and occurs at x = negative 2. This is the point negative 2, 16. The local minimum is negative 16 and occurs at x = 2. This is the point 2, negative 16.\" width=\"731\" height=\"467\" \/> <strong>Figure 4<\/strong>[\/caption]\r\n<p id=\"fs-id1165133316450\">To locate the local maxima and minima from a graph, we need to observe the graph to determine where the graph attains its highest and lowest points, respectively, within an open interval. Like the summit of a roller coaster, the graph of a function is higher at a local maximum than at nearby points on both sides. The graph will also be lower at a local minimum than at neighboring points. Figure 5\u00a0illustrates these ideas for a local maximum.<\/p>\r\n\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010554\/CNX_Precalc_Figure_01_03_0052.jpg\" alt=\"Graph of a polynomial that shows the increasing and decreasing intervals and local maximum.\" width=\"487\" height=\"295\" \/> <strong>Figure 5.<\/strong> Definition of a local maximum.[\/caption]\r\n<p id=\"eip-673\">These observations lead us to a formal definition of local extrema.<\/p>\r\n\r\n<div id=\"fs-id1165134169419\" class=\"note textbox\">\r\n<h3 class=\"title\">A General Note: Local Minima and Local Maxima<\/h3>\r\n<p id=\"fs-id1165134169426\">A function [latex]f[\/latex] is an <strong>increasing function<\/strong> on an open interval if [latex]f\\left(b\\right)&gt;f\\left(a\\right)[\/latex] for any two input values [latex]a[\/latex] and [latex]b[\/latex] in the given interval where [latex]b&gt;a[\/latex].<\/p>\r\n<p id=\"fs-id1165137668624\">A function [latex]f[\/latex] is a <strong>decreasing function<\/strong> on an open interval if [latex]f\\left(b\\right)&lt;f\\left(a\\right)[\/latex] for any two input values [latex]a[\/latex] and [latex]b[\/latex] in the given interval where [latex]b&gt;a[\/latex].<\/p>\r\n<p id=\"fs-id1165135389881\">A function [latex]f[\/latex] has a local maximum at [latex]x=b[\/latex] if there exists an interval [latex]\\left(a,c\\right)[\/latex] with [latex]a&lt;b&lt;c[\/latex] such that, for any [latex]x[\/latex] in the interval [latex]\\left(a,c\\right)[\/latex], [latex]f\\left(x\\right)\\le f\\left(b\\right)[\/latex]. Likewise, [latex]f[\/latex] has a local minimum at [latex]x=b[\/latex] if there exists an interval [latex]\\left(a,c\\right)[\/latex] with [latex]a&lt;b&lt;c[\/latex] such that, for any [latex]x[\/latex] in the interval [latex]\\left(a,c\\right)[\/latex], [latex]f\\left(x\\right)\\ge f\\left(b\\right)[\/latex].<\/p>\r\n\r\n<\/div>\r\n<div id=\"Example_01_03_07\" class=\"example\">\r\n<div id=\"fs-id1165134266716\" class=\"exercise\">\r\n<div id=\"fs-id1165134266718\" class=\"problem textbox shaded\">\r\n<h3>Example 7: Finding Increasing and Decreasing Intervals on a Graph<\/h3>\r\nGiven the function [latex]p\\left(t\\right)[\/latex] in the graph below, identify the intervals on which the function appears to be increasing.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010554\/CNX_Precalc_Figure_01_03_0062.jpg\" alt=\"Graph of a polynomial. As x gets large in the negative direction, the outputs of the function get large in the positive direction. As inputs approach 1, then the function value approaches a minimum of negative one. As x approaches 3, the values increase again and between 3 and 4 decrease one last time. As x gets large in the positive direction, the function values increase without bound.\" width=\"487\" height=\"295\" \/> <strong>Figure 6<\/strong>[\/caption]\r\n\r\n[reveal-answer q=\"331055\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"331055\"]\r\n<p id=\"fs-id1165133067197\">We see that the function is not constant on any interval. The function is increasing where it slants upward as we move to the right and decreasing where it slants downward as we move to the right. The function appears to be increasing from [latex]t=1[\/latex] to [latex]t=3[\/latex] and from [latex]t=4[\/latex] on.<\/p>\r\n<p id=\"fs-id1165135369127\">In <strong>interval notation<\/strong>, we would say the function appears to be increasing on the interval (1,3) and the interval [latex]\\left(4,\\infty \\right)[\/latex].<\/p>\r\n\r\n<h4>Analysis of the Solution<\/h4>\r\n<p id=\"fs-id1165134104021\">Notice in this example that we used open intervals (intervals that do not include the endpoints), because the function is neither increasing nor decreasing at [latex]t=1[\/latex] , [latex]t=3[\/latex] , and [latex]t=4[\/latex] . These points are the local extrema (two minima and a maximum).<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"Example_01_03_08\" class=\"example\">\r\n<div id=\"fs-id1165135261521\" class=\"exercise\">\r\n<div id=\"fs-id1165135261523\" class=\"problem textbox shaded\">\r\n<h3>Example 8: Finding Local Extrema from a Graph<\/h3>\r\n<p id=\"fs-id1165135261528\">Graph the function [latex]f\\left(x\\right)=\\frac{2}{x}+\\frac{x}{3}[\/latex]. Then use the graph to estimate the local extrema of the function and to determine the intervals on which the function is increasing.<\/p>\r\n[reveal-answer q=\"453777\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"453777\"]\r\n\r\nUsing technology, we find that the graph of the function looks like that in Figure 7. It appears there is a low point, or local minimum, between [latex]x=2[\/latex] and [latex]x=3[\/latex], and a mirror-image high point, or local maximum, somewhere between [latex]x=-3[\/latex] and [latex]x=-2[\/latex].\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010554\/CNX_Precalc_Figure_01_03_0072.jpg\" alt=\"Graph of a reciprocal function.\" width=\"487\" height=\"368\" \/> <b>Figure 7<\/b>[\/caption]\r\n<h4>Analysis of the Solution<\/h4>\r\nMost graphing calculators and graphing utilities can estimate the location of maxima and minima. Figure 7\u00a0provides screen images from two different technologies, showing the estimate for the local maximum and minimum.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010555\/CNX_Precalc_Figure_01_03_008ab2.jpg\" alt=\"Graph of the reciprocal function on a graphing calculator.\" width=\"975\" height=\"376\" \/> <b>Figure 8<\/b>[\/caption]\r\n<p id=\"fs-id1165134075625\">Based on these estimates, the function is increasing on the interval [latex]\\left(-\\infty ,-{2.449}\\right)[\/latex]\r\nand [latex]\\left(2.449\\text{,}\\infty \\right)[\/latex]. Notice that, while we expect the extrema to be symmetric, the two different technologies agree only up to four decimals due to the differing approximation algorithms used by each. (The exact location of the extrema is at [latex]\\pm \\sqrt{6}[\/latex], but determining this requires calculus.)<\/p>\r\n[\/hidden-answer]<b><\/b>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1165135640967\">Graph the function [latex]f\\left(x\\right)={x}^{3}-6{x}^{2}-15x+20[\/latex] to estimate the local extrema of the function. Use these to determine the intervals on which the function is increasing and decreasing.<\/p>\r\n[reveal-answer q=\"48622\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"48622\"]\r\n\r\nThe local maximum is 28 at\u00a0<em>x\u00a0<\/em>= -1\u00a0and the local minimum is -80 at\u00a0<em>x<\/em> = 5. The function is increasing on [latex]\\left(-\\infty ,-1\\right)\\cup \\left(5,\\infty \\right)[\/latex] and decreasing on [latex]\\left(-1,5\\right)[\/latex].\r\n\r\n<span id=\"fs-id1165134043615\">\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010555\/CNX_Precalc_Figure_01_03_0102.jpg\" alt=\"Graph of a polynomial with a local maximum at (-1, 28) and local minimum at (5, -80).\" width=\"487\" height=\"328\" \/><\/span>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\n[ohm_question hide_question_numbers=1]165724[\/ohm_question]\r\n\r\n<\/div>\r\n<div id=\"Example_01_03_09\" class=\"example\">\r\n<div id=\"fs-id1165135367558\" class=\"exercise\">\r\n<div id=\"fs-id1165137896103\" class=\"problem textbox shaded\">\r\n<h3>Example 9: Finding Local Maxima and Minima from a Graph<\/h3>\r\nFor the function [latex]f[\/latex] whose graph is shown in Figure 9, find all local maxima and minima.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010555\/CNX_Precalc_Figure_01_03_0112.jpg\" alt=\"Graph of a polynomial. The line curves down to x = negative 2 and up to x = 1.\" width=\"487\" height=\"368\" \/> <b>Figure 9<\/b>[\/caption]\r\n\r\n[reveal-answer q=\"209462\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"209462\"]\r\n<p id=\"fs-id1165135527085\">Observe the graph of [latex]f[\/latex]. The graph attains a local maximum at [latex]x=1[\/latex] because it is the highest point in an open interval around [latex]x=1[\/latex]. The local maximum is the [latex]y[\/latex] -coordinate at [latex]x=1[\/latex], which is [latex]2[\/latex].<\/p>\r\n<p id=\"fs-id1165134485672\">The graph attains a local minimum at [latex]\\text{ }x=-1\\text{ }[\/latex] because it is the lowest point in an open interval around [latex]x=-1[\/latex]. The local minimum is the <em>y<\/em>-coordinate at [latex]x=-1[\/latex], which is [latex]-2[\/latex].<\/p>\r\n[\/hidden-answer]<b><\/b>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<section id=\"fs-id1165134544960\">\r\n<h1 style=\"text-align: center;\"><span style=\"text-decoration: underline;\">Analyzing the Toolkit Functions for Increasing or Decreasing Intervals<\/span><\/h1>\r\nWe will now return to our toolkit functions and discuss their graphical behavior in the table\u00a0below.\r\n<table>\r\n<thead>\r\n<tr>\r\n<th>Function<\/th>\r\n<th>Increasing\/Decreasing<\/th>\r\n<th>Example<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>Constant Function\r\n\r\n[latex]f\\left(x\\right)={c}[\/latex]<\/td>\r\n<td>Neither increasing nor decreasing<\/td>\r\n<td>\u00a0<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012927\/Screen-Shot-2015-08-20-at-8.52.37-AM.png\"><img class=\"alignnone wp-image-12510 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012927\/Screen-Shot-2015-08-20-at-8.52.37-AM.png\" alt=\"\" width=\"143\" height=\"146\" \/><\/a><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Identity Function\r\n\r\n[latex]f\\left(x\\right)={x}[\/latex]<\/td>\r\n<td>\u00a0Increasing<\/td>\r\n<td><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012928\/Screen-Shot-2015-08-20-at-8.52.47-AM.png\"><img class=\"alignnone wp-image-12511 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012928\/Screen-Shot-2015-08-20-at-8.52.47-AM.png\" alt=\"\" width=\"143\" height=\"147\" \/><\/a><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Quadratic Function\r\n\r\n[latex]f\\left(x\\right)={x}^{2}[\/latex]<\/td>\r\n<td>Increasing on\u00a0[latex]\\left(0,\\infty\\right)[\/latex]\r\n\r\nDecreasing on\u00a0[latex]\\left(-\\infty,0\\right)[\/latex]\r\n\r\nMinimum at [latex]x=0[\/latex]<\/td>\r\n<td><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012928\/Screen-Shot-2015-08-20-at-8.52.54-AM.png\"><img class=\"alignnone size-full wp-image-12512\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012928\/Screen-Shot-2015-08-20-at-8.52.54-AM.png\" alt=\"\" width=\"138\" height=\"143\" \/><\/a><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Cubic Function\r\n\r\n[latex]f\\left(x\\right)={x}^{3}[\/latex]<\/td>\r\n<td>Increasing<\/td>\r\n<td><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012928\/Screen-Shot-2015-08-20-at-8.53.02-AM.png\"><img class=\"alignnone wp-image-12513 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012928\/Screen-Shot-2015-08-20-at-8.53.02-AM.png\" alt=\"\" width=\"137\" height=\"145\" \/><\/a><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\u00a0Reciprocal\r\n\r\n[latex]f\\left(x\\right)=\\frac{1}{x}[\/latex]<\/td>\r\n<td>Decreasing [latex]\\left(-\\infty,0\\right)\\cup\\left(0,\\infty\\right)[\/latex]<\/td>\r\n<td><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012928\/Screen-Shot-2015-08-20-at-8.53.09-AM.png\"><img class=\"alignnone wp-image-12514 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012928\/Screen-Shot-2015-08-20-at-8.53.09-AM.png\" alt=\"\" width=\"138\" height=\"146\" \/><\/a><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Reciprocal Squared\r\n\r\n[latex]f\\left(x\\right)=\\frac{1}{{x}^{2}}[\/latex]<\/td>\r\n<td>Increasing on\u00a0[latex]\\left(-\\infty,0\\right)[\/latex]\r\n\r\nDecreasing on\u00a0[latex]\\left(0,\\infty\\right)[\/latex]<\/td>\r\n<td><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012928\/Screen-Shot-2015-08-20-at-8.53.16-AM.png\"><img class=\"alignnone size-full wp-image-12515\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012928\/Screen-Shot-2015-08-20-at-8.53.16-AM.png\" alt=\"\" width=\"134\" height=\"145\" \/><\/a><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Cube Root\r\n\r\n[latex]f\\left(x\\right)=\\sqrt[3]{x}[\/latex]\r\n\r\n&nbsp;<\/td>\r\n<td>Increasing<\/td>\r\n<td>\u00a0<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012929\/Screen-Shot-2015-08-20-at-8.53.26-AM.png\"><img class=\"alignnone size-full wp-image-12516\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012929\/Screen-Shot-2015-08-20-at-8.53.26-AM.png\" alt=\"Screen Shot 2015-08-20 at 8.53.26 AM\" width=\"140\" height=\"147\" \/><\/a><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Square Root\r\n\r\n[latex]f\\left(x\\right)=\\sqrt{x}[\/latex]<\/td>\r\n<td>Increasing on [latex]\\left(0,\\infty\\right)[\/latex]<\/td>\r\n<td>\u00a0<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012929\/Screen-Shot-2015-08-20-at-8.53.33-AM.png\"><img class=\"alignnone size-full wp-image-12517\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012929\/Screen-Shot-2015-08-20-at-8.53.33-AM.png\" alt=\"\" width=\"138\" height=\"142\" \/><\/a><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Absolute Value\r\n\r\n[latex]f\\left(x\\right)=|x|[\/latex]<\/td>\r\n<td>Increasing on [latex]\\left(0,\\infty\\right)[\/latex]\r\n\r\nDecreasing on\u00a0[latex]\\left(-\\infty,0\\right)[\/latex]<\/td>\r\n<td>\u00a0<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012929\/Screen-Shot-2015-08-20-at-8.53.40-AM.png\"><img class=\"alignnone wp-image-12518 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012929\/Screen-Shot-2015-08-20-at-8.53.40-AM.png\" alt=\"\" width=\"135\" height=\"143\" \/><\/a><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h2>Use\u00a0A Graph to Locate the Absolute Maximum and Absolute Minimum<\/h2>\r\nThere is a difference between locating the highest and lowest points on a graph in a region around an open interval (locally) and locating the highest and lowest points on the graph for the entire domain. The [latex]y\\text{-}[\/latex] coordinates (output) at the highest and lowest points are called the <strong>absolute maximum <\/strong>and<strong> absolute minimum<\/strong>, respectively.\r\n\r\nTo locate absolute maxima and minima from a graph, we need to observe the graph to determine where the graph attains it highest and lowest points on the domain of the function. See Figure 10.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010557\/CNX_Precalc_Figure_01_03_0152.jpg\" alt=\"Graph of a segment of a parabola with an absolute minimum at (0, -2) and absolute maximum at (2, 2).\" width=\"487\" height=\"323\" \/> <b>Figure 10<\/b>[\/caption]\r\n<p id=\"fs-id1165137692066\">Not every function has an absolute maximum or minimum value. The toolkit function [latex]f\\left(x\\right)={x}^{3}[\/latex] is one such function.<\/p>\r\n\r\n<div id=\"fs-id1165135251290\" class=\"note textbox\">\r\n<h3 class=\"title\">A General Note: Absolute Maxima and Minima<\/h3>\r\n<p id=\"fs-id1165132939786\">The <strong>absolute maximum<\/strong> of [latex]f[\/latex] at [latex]x=c[\/latex] is [latex]f\\left(c\\right)[\/latex] where [latex]f\\left(c\\right)\\ge f\\left(x\\right)[\/latex] for all [latex]x[\/latex] in the domain of [latex]f[\/latex].<\/p>\r\n<p id=\"fs-id1165137932685\">The <strong>absolute minimum<\/strong> of [latex]f[\/latex] at [latex]x=d[\/latex] is [latex]f\\left(d\\right)[\/latex] where [latex]f\\left(d\\right)\\le f\\left(x\\right)[\/latex] for all [latex]x[\/latex] in the domain of [latex]f[\/latex].<\/p>\r\n\r\n<\/div>\r\n<div id=\"Example_01_03_10\" class=\"example\">\r\n<div id=\"fs-id1165134047533\" class=\"exercise\">\r\n<div id=\"fs-id1165134047535\" class=\"problem textbox shaded\">\r\n<h3>Example 10: Finding Absolute Maxima and Minima from a Graph<\/h3>\r\nFor the function [latex]f[\/latex] shown in Figure 11, find all absolute maxima and minima.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010557\/CNX_Precalc_Figure_01_03_0132.jpg\" alt=\"Graph of a polynomial.\" width=\"487\" height=\"403\" \/> <b>Figure 11<\/b>[\/caption]\r\n\r\n[reveal-answer q=\"502428\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"502428\"]\r\n<p id=\"fs-id1165135532371\">Observe the graph of [latex]f[\/latex]. The graph attains an absolute maximum in two locations, [latex]x=-2[\/latex] and [latex]x=2[\/latex], because at these locations, the graph attains its highest point on the domain of the function. The absolute maximum is the <em>y<\/em>-coordinate at [latex]x=-2[\/latex] and [latex]x=2[\/latex], which is [latex]16[\/latex].<\/p>\r\n<p id=\"fs-id1165137863670\">The graph attains an absolute minimum at [latex]x=3[\/latex], because it is the lowest point on the domain of the function\u2019s graph. The absolute minimum is the <em>y<\/em>-coordinate at [latex]x=3[\/latex], which is [latex]-10[\/latex].<\/p>\r\n[\/hidden-answer]<b><\/b>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135708033\" class=\"note precalculus media\"><section id=\"fs-id1165135541564\" class=\"key-equations\">\r\n<h2>Key Equations<\/h2>\r\n<table id=\"eip-id1165135358784\" summary=\"..\"><colgroup> <col \/> <col \/><\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td>Average rate of change<\/td>\r\n<td>[latex]\\dfrac{\\Delta y}{\\Delta x}=\\dfrac{f\\left({x}_{2}\\right)-f\\left({x}_{1}\\right)}{{x}_{2}-{x}_{1}}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/section><section id=\"fs-id1165135481945\" class=\"key-concepts\">\r\n<h2>Key Concepts<\/h2>\r\n<ul id=\"fs-id1165135481952\">\r\n \t<li>A rate of change relates a change in an output quantity to a change in an input quantity. The average rate of change is determined using only the beginning and ending data.<\/li>\r\n \t<li>Identifying points that mark the interval on a graph can be used to find the average rate of change.<\/li>\r\n \t<li>Comparing pairs of input and output values in a table can also be used to find the average rate of change.<\/li>\r\n \t<li>An average rate of change can also be computed by determining the function values at the endpoints of an interval described by a formula.<\/li>\r\n \t<li>The average rate of change can sometimes be determined as an expression.<\/li>\r\n \t<li>A function is increasing where its rate of change is positive and decreasing where its rate of change is negative.<\/li>\r\n \t<li>A local maximum is where a function changes from increasing to decreasing and has an output value larger (more positive or less negative) than output values at neighboring input values.<\/li>\r\n \t<li>A local minimum is where the function changes from decreasing to increasing (as the input increases) and has an output value smaller (more negative or less positive) than output values at neighboring input values.<\/li>\r\n \t<li>Minima and maxima are also called extrema.<\/li>\r\n \t<li>We can find local extrema from a graph.<\/li>\r\n \t<li>The highest and lowest points on a graph indicate the maxima and minima.<\/li>\r\n<\/ul>\r\n<div style=\"line-height: 1.5;\">\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1165133052903\" class=\"definition\">\r\n \t<dt><strong>absolute maximum<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165133052908\">the greatest value of a function over an interval<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165133052911\" class=\"definition\">\r\n \t<dt><strong>absolute minimum<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165133052916\">the lowest value of a function over an interval<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165133052921\" class=\"definition\">\r\n \t<dt><strong>average rate of change<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165133052926\">the difference in the output values of a function found for two values of the input divided by the difference between the inputs<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165135264639\" class=\"definition\">\r\n \t<dt><strong>decreasing function<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165135264645\">a function is decreasing in some open interval if [latex]f\\left(b\\right)&lt;f\\left(a\\right)[\/latex] for any two input values [latex]a[\/latex] and [latex]b[\/latex] in the given interval where [latex]b&gt;a[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165135639824\" class=\"definition\">\r\n \t<dt><strong>increasing function<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165135639829\">a function is increasing in some open interval if [latex]f\\left(b\\right)&gt;f\\left(a\\right)[\/latex] for any two input values [latex]a[\/latex] and [latex]b[\/latex] in the given interval where [latex]b&gt;a[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165135536408\" class=\"definition\">\r\n \t<dt><strong>local extrema<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165135536413\">collectively, all of a function's local maxima and minima<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165135536416\" class=\"definition\">\r\n \t<dt><strong>local maximum<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165135412035\">a value of the input where a function changes from increasing to decreasing as the input value increases.<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165135412040\" class=\"definition\">\r\n \t<dt><strong>local minimum<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165135412046\">a value of the input where a function changes from decreasing to increasing as the input value increases.<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165135412050\" class=\"definition\">\r\n \t<dt><strong>rate of change<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165135412054\">the change of an output quantity relative to the change of the input quantity<\/dd>\r\n<\/dl>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><\/div>\r\n<\/div>","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Find the average rate of change of a function.<\/li>\n<li>Use a graph to determine where a function is increasing, decreasing, or constant.<\/li>\n<li>Use a graph to locate local and absolute maxima and local minima.<\/li>\n<\/ul>\n<\/div>\n<p id=\"fs-id1165135194500\">Gasoline costs have experienced some wild fluctuations over the last several decades. The table below<a class=\"footnote\" title=\"http:\/\/www.eia.gov\/totalenergy\/data\/annual\/showtext.cfm?t=ptb0524. Accessed 3\/5\/2014.\" id=\"return-footnote-13679-1\" href=\"#footnote-13679-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a>\u00a0lists the average cost, in dollars, of a gallon of gasoline for the years 2005\u20132012. The cost of gasoline can be considered as a function of year.<\/p>\n<table summary=\"Two rows and nine columns. The first row is labeled,\">\n<colgroup>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/><\/colgroup>\n<tbody>\n<tr>\n<td><strong>[latex]y[\/latex]<\/strong><\/td>\n<td>2005<\/td>\n<td>2006<\/td>\n<td>2007<\/td>\n<td>2008<\/td>\n<td>2009<\/td>\n<td>2010<\/td>\n<td>2011<\/td>\n<td>2012<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]C\\left(y\\right)[\/latex]<\/strong><\/td>\n<td>2.31<\/td>\n<td>2.62<\/td>\n<td>2.84<\/td>\n<td>3.30<\/td>\n<td>2.41<\/td>\n<td>2.84<\/td>\n<td>3.58<\/td>\n<td>3.68<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165133097252\">If we were interested only in how the gasoline prices changed between 2005 and 2012, we could compute that the cost per gallon had increased from $2.31 to $3.68, an increase of $1.37. While this is interesting, it might be more useful to look at how much the price changed <em>per year<\/em>. In this section, we will investigate changes such as these.<\/p>\n<h2>Finding the Average Rate of Change of a Function<\/h2>\n<p id=\"fs-id1165137834011\">The price change per year is a <strong>rate of change<\/strong> because it describes how an output quantity changes relative to the change in the input quantity. We can see that the price of gasoline in\u00a0the table above\u00a0did not change by the same amount each year, so the rate of change was not constant. If we use only the beginning and ending data, we would be finding the <strong>average rate of change<\/strong> over the specified period of time. To find the average rate of change, we divide the change in the output value by the change in the input value.<\/p>\n<div id=\"fs-id1165135452482\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{align}\\text{Average rate of change}&=\\frac{\\text{Change in output}}{\\text{Change in input}} \\\\[1mm] &=\\frac{\\Delta y}{\\Delta x} \\\\[1mm] &= \\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}} \\\\[1mm] &= \\frac{f\\left({x}_{2}\\right)-f\\left({x}_{1}\\right)}{{x}_{2}-{x}_{1}}\\end{align}[\/latex]<\/div>\n<p id=\"fs-id1165135471272\">The Greek letter [latex]\\Delta[\/latex] (delta) signifies the change in a quantity; we read the ratio as &#8220;delta-<em>y<\/em> over delta-<em>x<\/em>&#8221; or &#8220;the change in [latex]y[\/latex] divided by the change in [latex]x[\/latex].&#8221; Occasionally we write [latex]\\Delta f[\/latex] instead of [latex]\\Delta y[\/latex], which still represents the change in the function\u2019s output value resulting from a change to its input value. It does not mean we are changing the function into some other function.<\/p>\n<p id=\"fs-id1165137539940\">In our example, the gasoline price increased by $1.37 from 2005 to 2012. Over 7 years, the average rate of change was<\/p>\n<div id=\"fs-id1165137526960\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\frac{\\Delta y}{\\Delta x}=\\frac{{1.37}}{\\text{7 years}}\\approx 0.196\\text{ dollars per year}[\/latex]<\/div>\n<p id=\"fs-id1165137418924\">On average, the price of gas increased by about 19.6\u00a2 each year.<\/p>\n<p id=\"fs-id1165135397217\">Other examples of rates of change include:<\/p>\n<ul id=\"fs-id1165137424067\">\n<li>A population of rats increasing by 40 rats per week<\/li>\n<li>A car traveling 68 miles per hour (distance traveled changes by 68 miles each hour as time passes)<\/li>\n<li>A car driving 27 miles per gallon (distance traveled changes by 27 miles for each gallon)<\/li>\n<li>The current through an electrical circuit increasing by 0.125 amperes for every volt of increased voltage<\/li>\n<li>The amount of money in a college account decreasing by $4,000 per quarter<\/li>\n<\/ul>\n<div class=\"textbox\">\n<h3 class=\"title\">A General Note: Rate of Change<\/h3>\n<p id=\"fs-id1165137780744\">A rate of change describes how an output quantity changes relative to the change in the input quantity. The units on a rate of change are &#8220;output units per input units.&#8221;<\/p>\n<p id=\"fs-id1165137544638\">The average rate of change between two input values is the total change of the function values (output values) divided by the change in the input values.<\/p>\n<div id=\"fs-id1165134060431\" class=\"equation\" style=\"text-align: center;\">[latex]\\frac{\\Delta y}{\\Delta x}=\\frac{f\\left({x}_{2}\\right)-f\\left({x}_{1}\\right)}{{x}_{2}-{x}_{1}}[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165135530407\" class=\"note precalculus howto textbox\">\n<h3 id=\"fs-id1165137762240\">How To: Given the value of a function at different points, calculate the average rate of change of a function for the interval between two values [latex]{x}_{1}[\/latex] and [latex]{x}_{2}[\/latex].<\/h3>\n<ol id=\"fs-id1165137442714\">\n<li>Calculate the difference [latex]{y}_{2}-{y}_{1}=\\Delta y[\/latex].<\/li>\n<li>Calculate the difference [latex]{x}_{2}-{x}_{1}=\\Delta x[\/latex].<\/li>\n<li>Find the ratio [latex]\\frac{\\Delta y}{\\Delta x}[\/latex].<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_01_03_01\" class=\"example\">\n<div id=\"fs-id1165135485962\" class=\"exercise\">\n<div id=\"fs-id1165137464225\" class=\"problem textbox shaded\">\n<h3>Example 1: Computing an Average Rate of Change<\/h3>\n<p id=\"fs-id1165137603118\">Using the data in the table below, find the average rate of change of the price of gasoline between 2007 and 2009.<\/p>\n<table summary=\"Two rows and nine columns. The first row is labeled,\">\n<colgroup>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/><\/colgroup>\n<tbody>\n<tr>\n<td><strong>[latex]y[\/latex]<\/strong><\/td>\n<td>2005<\/td>\n<td>2006<\/td>\n<td>2007<\/td>\n<td>2008<\/td>\n<td>2009<\/td>\n<td>2010<\/td>\n<td>2011<\/td>\n<td>2012<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]C\\left(y\\right)[\/latex]<\/strong><\/td>\n<td>2.31<\/td>\n<td>2.62<\/td>\n<td>2.84<\/td>\n<td>3.30<\/td>\n<td>2.41<\/td>\n<td>2.84<\/td>\n<td>3.58<\/td>\n<td>3.68<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q562005\">Show Solution<\/span><\/p>\n<div id=\"q562005\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135209401\">In 2007, the price of gasoline was $2.84. In 2009, the cost was $2.41. The average rate of change is<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}\\frac{\\Delta y}{\\Delta x}&=\\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}} \\\\[1mm] &=\\frac{2.41-2.84}{2009 - 2007} \\\\[1mm] &=\\frac{-0.43}{2\\text{ years}} \\\\[1mm] &={-0.22}\\text{ per year}\\end{align}[\/latex]<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p id=\"fs-id1165137784092\">Note that a decrease is expressed by a negative change or &#8220;negative increase.&#8221; A rate of change is negative when the output decreases as the input increases or when the output increases as the input decreases.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>The following video provides another example of how to find the average rate of change between two points from a table of values.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex:  Find the Average Rate of Change From a Table - Temperatures\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/iJ_0nPUUlOg?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1165135160759\">Using the data in the table below,\u00a0find the average rate of change between 2005 and 2010.<\/p>\n<table summary=\"Two rows and nine columns. The first row is labeled,\">\n<tbody>\n<tr>\n<td><strong>[latex]y[\/latex]<\/strong><\/td>\n<td>2005<\/td>\n<td>2006<\/td>\n<td>2007<\/td>\n<td>2008<\/td>\n<td>2009<\/td>\n<td>2010<\/td>\n<td>2011<\/td>\n<td>2012<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]C\\left(y\\right)[\/latex]<\/strong><\/td>\n<td>2.31<\/td>\n<td>2.62<\/td>\n<td>2.84<\/td>\n<td>3.30<\/td>\n<td>2.41<\/td>\n<td>2.84<\/td>\n<td>3.58<\/td>\n<td>3.68<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q175600\">Show Solution<\/span><\/p>\n<div id=\"q175600\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\dfrac{$2.84-$2.31}{5\\text{ years}}=\\dfrac{$0.53}{5\\text{ years}}=$0.106[\/latex] per year.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_01_03_02\" class=\"example\">\n<div id=\"fs-id1165137851963\" class=\"exercise\">\n<div id=\"fs-id1165137437853\" class=\"problem textbox shaded\">\n<h3>Example 2: Computing Average Rate of Change from a Graph<\/h3>\n<p>Given the function [latex]g\\left(t\\right)[\/latex] shown in Figure 1, find the average rate of change on the interval [latex]\\left[-1,2\\right][\/latex].<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010553\/CNX_Precalc_Figure_01_03_0012.jpg\" alt=\"Graph of a parabola.\" width=\"487\" height=\"295\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 1<\/b><\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q806109\">Show Solution<\/span><\/p>\n<div id=\"q806109\" class=\"hidden-answer\" style=\"display: none\">\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010553\/CNX_Precalc_Figure_01_03_0022.jpg\" alt=\"Graph of a parabola with a line from points (-1, 4) and (2, 1) to show the changes for g(t) and t.\" width=\"487\" height=\"296\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 2<\/b><\/p>\n<\/div>\n<p>At [latex]t=-1[\/latex], the graph\u00a0shows [latex]g\\left(-1\\right)=4[\/latex]. At [latex]t=2[\/latex], the graph shows [latex]g\\left(2\\right)=1[\/latex].<span id=\"fs-id1165137387448\"><br \/>\n<\/span><\/p>\n<p id=\"fs-id1165137591169\">The horizontal change [latex]\\Delta t=3[\/latex] is shown by the red arrow, and the vertical change [latex]\\Delta g\\left(t\\right)=-3[\/latex] is shown by the turquoise arrow. The output changes by \u20133 while the input changes by 3, giving an average rate of change of<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{1 - 4}{2-\\left(-1\\right)}=\\frac{-3}{3}=-1[\/latex]<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p id=\"fs-id1165135538482\">Note that the order we choose is very important. If, for example, we use [latex]\\frac{{y}_{2}-{y}_{1}}{{x}_{1}-{x}_{2}}[\/latex], we will not get the correct answer. Decide which point will be 1 and which point will be 2, and keep the coordinates fixed as [latex]\\left({x}_{1},{y}_{1}\\right)[\/latex] and [latex]\\left({x}_{2},{y}_{2}\\right)[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_01_03_03\" class=\"example\">\n<div id=\"fs-id1165135536188\" class=\"exercise\">\n<div id=\"fs-id1165137835656\" class=\"problem textbox shaded\">\n<h3>Example 3: Computing Average Rate of Change from a Table<\/h3>\n<p id=\"fs-id1165135515898\">After picking up a friend who lives 10 miles away, Anna records her distance from home over time. The values are shown in the table below.\u00a0Find her average speed over the first 6 hours.<\/p>\n<table id=\"Table_01_03_02\" summary=\"Two rows and nine columns. The first row is labeled,\">\n<colgroup>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/><\/colgroup>\n<tbody>\n<tr>\n<td><strong><em>t<\/em> (hours)<\/strong><\/td>\n<td>0<\/td>\n<td>1<\/td>\n<td>2<\/td>\n<td>3<\/td>\n<td>4<\/td>\n<td>5<\/td>\n<td>6<\/td>\n<td>7<\/td>\n<\/tr>\n<tr>\n<td><strong><em>D<\/em>(<em>t<\/em>) (miles)<\/strong><\/td>\n<td>10<\/td>\n<td>55<\/td>\n<td>90<\/td>\n<td>153<\/td>\n<td>214<\/td>\n<td>240<\/td>\n<td>292<\/td>\n<td>300<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q566859\">Show Solution<\/span><\/p>\n<div id=\"q566859\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137891478\">Here, the average speed is the average rate of change. She traveled 282 miles in 6 hours, for an average speed of<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{292 - 10}{6 - 0} =\\frac{282}{6} =47[\/latex]<\/p>\n<p id=\"fs-id1165135400200\">The average speed is 47 miles per hour.<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p id=\"fs-id1165137731074\">Because the speed is not constant, the average speed depends on the interval chosen. For the interval [2,3], the average speed is 63 miles per hour.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_01_03_04\" class=\"example\">\n<div id=\"fs-id1165135353057\" class=\"exercise\">\n<div id=\"fs-id1165135383644\" class=\"problem textbox shaded\">\n<h3>Example 4: Computing Average Rate of Change for a Function Expressed as a Formula<\/h3>\n<p id=\"fs-id1165131958324\">Compute the average rate of change of [latex]f\\left(x\\right)={x}^{2}-\\frac{1}{x}[\/latex] on the interval [latex]\\text{[2,}\\text{4].}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q222718\">Show Solution<\/span><\/p>\n<div id=\"q222718\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137595441\">We can start by computing the function values at each <strong>endpoint<\/strong> of the interval.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}f\\left(2\\right)&={2}^{2}-\\frac{1}{2} &&& f\\left(4\\right)&={4}^{2}-\\frac{1}{4} \\\\ &=4-\\frac{1}{2} &&&& =16-{1}{4} \\\\ &=\\frac{7}{2} &&&& =\\frac{63}{4} \\end{align}[\/latex]<\/p>\n<p id=\"fs-id1165137427523\">Now we compute the average rate of change.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}\\text{Average rate of change}&=\\frac{f\\left(4\\right)-f\\left(2\\right)}{4 - 2} \\\\[1mm] &=\\frac{\\frac{63}{4}-\\frac{7}{2}}{4 - 2} \\\\[1mm] &=\\frac{\\frac{49}{4}}{2} \\\\[1mm] &=\\frac{49}{8} \\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>The following video provides another example of finding the average rate of change of a function given a formula and an interval.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Ex:  Find the Average Rate of Change Given a Function Rule\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/g93QEKJXeu4?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1165137832324\">Find the average rate of change of [latex]f\\left(x\\right)=x - 2\\sqrt{x}[\/latex] on the interval [latex]\\left[1,9\\right][\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q191250\">Show Solution<\/span><\/p>\n<div id=\"q191250\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\frac{1}{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm165703\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=165703&theme=oea&iframe_resize_id=ohm165703\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div id=\"Example_01_03_05\" class=\"example\">\n<div id=\"fs-id1165137772170\" class=\"exercise\">\n<div id=\"fs-id1165137772173\" class=\"problem textbox shaded\">\n<h3>Example 5: Finding the Average Rate of Change of a Force<\/h3>\n<p id=\"fs-id1165135443718\">The <strong>electrostatic force<\/strong> [latex]F[\/latex], measured in newtons, between two charged particles can be related to the distance between the particles [latex]d[\/latex], in centimeters, by the formula [latex]F\\left(d\\right)=\\frac{2}{{d}^{2}}[\/latex]. Find the average rate of change of force if the distance between the particles is increased from 2 cm to 6 cm.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q271117\">Show Solution<\/span><\/p>\n<div id=\"q271117\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137770364\">We are computing the average rate of change of [latex]F\\left(d\\right)=\\frac{2}{{d}^{2}}[\/latex] on the interval [latex]\\left[2,6\\right][\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}\\text{Average rate of change }&=\\frac{F\\left(6\\right)-F\\left(2\\right)}{6 - 2} \\\\[1mm] &=\\frac{\\frac{2}{{6}^{2}}-\\frac{2}{{2}^{2}}}{6 - 2} && \\text{Simplify}. \\\\[1mm] &=\\frac{\\frac{2}{36}-\\frac{2}{4}}{4} \\\\[1mm] &=\\frac{-\\frac{16}{36}}{4} &&\\text{Combine numerator terms}. \\\\[1mm] &=-\\frac{1}{9}&&\\text{Simplify}\\end{align}[\/latex]<\/p>\n<p id=\"fs-id1165135543242\">The average rate of change is [latex]-\\frac{1}{9}[\/latex] newton per centimeter.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_01_03_06\" class=\"example\">\n<div id=\"fs-id1165135174952\" class=\"exercise\">\n<div id=\"fs-id1165135174954\" class=\"problem textbox shaded\">\n<h3>Example 6: Finding an Average Rate of Change as an Expression<\/h3>\n<p id=\"fs-id1165135155397\">Find the average rate of change of [latex]g\\left(t\\right)={t}^{2}+3t+1[\/latex] on the interval [latex]\\left[0,a\\right][\/latex]. The answer will be an expression involving [latex]a[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q138559\">Show Solution<\/span><\/p>\n<div id=\"q138559\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137418913\">We use the average rate of change formula.<\/p>\n<p style=\"text-align: center;\">\u200b[latex]\\begin{align}\\text{Average rate of change}&=\\frac{g\\left(a\\right)-g\\left(0\\right)}{a - 0}&&\\text{Evaluate} \\\\[1mm] &\u200b=\\frac{\\left({a}^{2}+3a+1\\right)-\\left({0}^{2}+3\\left(0\\right)+1\\right)}{a - 0}&&\\text{Simplify}\u200b \\\\[1mm] &=\\frac{{a}^{2}+3a+1 - 1}{a}&&\\text{Simplify and factor}\u200b \\\\[1mm] &=\\frac{a\\left(a+3\\right)}{a}&&\\text{Divide by the common factor }a\u200b \\\\[1mm] &=a+3 \\end{align}[\/latex]<\/p>\n<p id=\"fs-id1165133316469\">This result tells us the average rate of change in terms of [latex]a[\/latex] between [latex]t=0[\/latex] and any other point [latex]t=a[\/latex]. For example, on the interval [latex]\\left[0,5\\right][\/latex], the average rate of change would be [latex]5+3=8[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1165134149846\">Find the average rate of change of [latex]f\\left(x\\right)={x}^{2}+2x - 8[\/latex] on the interval [latex]\\left[5,a\\right][\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q944512\">Show Solution<\/span><\/p>\n<div id=\"q944512\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]a+7[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137784644\">As part of exploring how functions change, we can identify intervals over which the function is changing in specific ways. We say that a function is increasing on an interval if the function values increase as the input values increase within that interval. Similarly, a function is decreasing on an interval if the function values decrease as the input values increase over that interval. The average rate of change of an increasing function is positive, and the average rate of change of a decreasing function is negative. Figure 3\u00a0shows examples of increasing and decreasing intervals on a function.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010554\/CNX_Precalc_Figure_01_03_0042.jpg\" alt=\"Graph of a polynomial that shows the increasing and decreasing intervals and local maximum and minimum.\" width=\"487\" height=\"518\" \/><\/p>\n<p style=\"text-align: center;\"><strong>Figure 3.<\/strong> The function [latex]f\\left(x\\right)={x}^{3}-12x[\/latex] is increasing on [latex]\\left(-\\infty \\text{,}-\\text{2}\\right){{\\cup }^{\\text{ }}}^{\\text{ }}\\left(2,\\infty \\right)[\/latex] and is decreasing on [latex]\\left(-2\\text{,}2\\right)[\/latex].<\/p>\n<p>This video further explains how to find where a function is increasing or decreasing.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Determine Where a Function is Increasing and Decreasing\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/78b4HOMVcKM?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p id=\"fs-id1165134272749\">While some functions are increasing (or decreasing) over their entire domain, many others are not. A value of the input where a function changes from increasing to decreasing (as we go from left to right, that is, as the input variable increases) is called a <strong>local maximum<\/strong>. If a function has more than one, we say it has local maxima. Similarly, a value of the input where a function changes from decreasing to increasing as the input variable increases is called a <strong>local minimum<\/strong>. The plural form is &#8220;local minima.&#8221; Together, local maxima and minima are called <strong>local extrema<\/strong>, or local extreme values, of the function. (The singular form is &#8220;extremum.&#8221;) Often, the term <em>local<\/em> is replaced by the term <em>relative<\/em>. In this text, we will use the term <em>local<\/em>.<\/p>\n<p id=\"fs-id1165134547216\">Clearly, a function is neither increasing nor decreasing on an interval where it is constant. A function is also neither increasing nor decreasing at extrema. Note that we have to speak of <em>local<\/em> extrema, because any given local extremum as defined here is not necessarily the highest maximum or lowest minimum in the function\u2019s entire domain.<\/p>\n<p>For the function in Figure 4, the local maximum is 16, and it occurs at [latex]x=-2[\/latex]. The local minimum is [latex]-16[\/latex] and it occurs at [latex]x=2[\/latex].<\/p>\n<div style=\"width: 741px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010554\/CNX_Precalc_Figure_01_03_0142.jpg\" alt=\"Graph of a polynomial that shows the increasing and decreasing intervals and local maximum and minimum. The local maximum is 16 and occurs at x = negative 2. This is the point negative 2, 16. The local minimum is negative 16 and occurs at x = 2. This is the point 2, negative 16.\" width=\"731\" height=\"467\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 4<\/strong><\/p>\n<\/div>\n<p id=\"fs-id1165133316450\">To locate the local maxima and minima from a graph, we need to observe the graph to determine where the graph attains its highest and lowest points, respectively, within an open interval. Like the summit of a roller coaster, the graph of a function is higher at a local maximum than at nearby points on both sides. The graph will also be lower at a local minimum than at neighboring points. Figure 5\u00a0illustrates these ideas for a local maximum.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010554\/CNX_Precalc_Figure_01_03_0052.jpg\" alt=\"Graph of a polynomial that shows the increasing and decreasing intervals and local maximum.\" width=\"487\" height=\"295\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 5.<\/strong> Definition of a local maximum.<\/p>\n<\/div>\n<p id=\"eip-673\">These observations lead us to a formal definition of local extrema.<\/p>\n<div id=\"fs-id1165134169419\" class=\"note textbox\">\n<h3 class=\"title\">A General Note: Local Minima and Local Maxima<\/h3>\n<p id=\"fs-id1165134169426\">A function [latex]f[\/latex] is an <strong>increasing function<\/strong> on an open interval if [latex]f\\left(b\\right)>f\\left(a\\right)[\/latex] for any two input values [latex]a[\/latex] and [latex]b[\/latex] in the given interval where [latex]b>a[\/latex].<\/p>\n<p id=\"fs-id1165137668624\">A function [latex]f[\/latex] is a <strong>decreasing function<\/strong> on an open interval if [latex]f\\left(b\\right)<f\\left(a\\right)[\/latex] for any two input values [latex]a[\/latex] and [latex]b[\/latex] in the given interval where [latex]b>a[\/latex].<\/p>\n<p id=\"fs-id1165135389881\">A function [latex]f[\/latex] has a local maximum at [latex]x=b[\/latex] if there exists an interval [latex]\\left(a,c\\right)[\/latex] with [latex]a<b<c[\/latex] such that, for any [latex]x[\/latex] in the interval [latex]\\left(a,c\\right)[\/latex], [latex]f\\left(x\\right)\\le f\\left(b\\right)[\/latex]. Likewise, [latex]f[\/latex] has a local minimum at [latex]x=b[\/latex] if there exists an interval [latex]\\left(a,c\\right)[\/latex] with [latex]a<b<c[\/latex] such that, for any [latex]x[\/latex] in the interval [latex]\\left(a,c\\right)[\/latex], [latex]f\\left(x\\right)\\ge f\\left(b\\right)[\/latex].<\/p>\n<\/div>\n<div id=\"Example_01_03_07\" class=\"example\">\n<div id=\"fs-id1165134266716\" class=\"exercise\">\n<div id=\"fs-id1165134266718\" class=\"problem textbox shaded\">\n<h3>Example 7: Finding Increasing and Decreasing Intervals on a Graph<\/h3>\n<p>Given the function [latex]p\\left(t\\right)[\/latex] in the graph below, identify the intervals on which the function appears to be increasing.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010554\/CNX_Precalc_Figure_01_03_0062.jpg\" alt=\"Graph of a polynomial. As x gets large in the negative direction, the outputs of the function get large in the positive direction. As inputs approach 1, then the function value approaches a minimum of negative one. As x approaches 3, the values increase again and between 3 and 4 decrease one last time. As x gets large in the positive direction, the function values increase without bound.\" width=\"487\" height=\"295\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 6<\/strong><\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q331055\">Show Solution<\/span><\/p>\n<div id=\"q331055\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165133067197\">We see that the function is not constant on any interval. The function is increasing where it slants upward as we move to the right and decreasing where it slants downward as we move to the right. The function appears to be increasing from [latex]t=1[\/latex] to [latex]t=3[\/latex] and from [latex]t=4[\/latex] on.<\/p>\n<p id=\"fs-id1165135369127\">In <strong>interval notation<\/strong>, we would say the function appears to be increasing on the interval (1,3) and the interval [latex]\\left(4,\\infty \\right)[\/latex].<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p id=\"fs-id1165134104021\">Notice in this example that we used open intervals (intervals that do not include the endpoints), because the function is neither increasing nor decreasing at [latex]t=1[\/latex] , [latex]t=3[\/latex] , and [latex]t=4[\/latex] . These points are the local extrema (two minima and a maximum).<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_01_03_08\" class=\"example\">\n<div id=\"fs-id1165135261521\" class=\"exercise\">\n<div id=\"fs-id1165135261523\" class=\"problem textbox shaded\">\n<h3>Example 8: Finding Local Extrema from a Graph<\/h3>\n<p id=\"fs-id1165135261528\">Graph the function [latex]f\\left(x\\right)=\\frac{2}{x}+\\frac{x}{3}[\/latex]. Then use the graph to estimate the local extrema of the function and to determine the intervals on which the function is increasing.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q453777\">Show Solution<\/span><\/p>\n<div id=\"q453777\" class=\"hidden-answer\" style=\"display: none\">\n<p>Using technology, we find that the graph of the function looks like that in Figure 7. It appears there is a low point, or local minimum, between [latex]x=2[\/latex] and [latex]x=3[\/latex], and a mirror-image high point, or local maximum, somewhere between [latex]x=-3[\/latex] and [latex]x=-2[\/latex].<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010554\/CNX_Precalc_Figure_01_03_0072.jpg\" alt=\"Graph of a reciprocal function.\" width=\"487\" height=\"368\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 7<\/b><\/p>\n<\/div>\n<h4>Analysis of the Solution<\/h4>\n<p>Most graphing calculators and graphing utilities can estimate the location of maxima and minima. Figure 7\u00a0provides screen images from two different technologies, showing the estimate for the local maximum and minimum.<\/p>\n<div style=\"width: 985px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010555\/CNX_Precalc_Figure_01_03_008ab2.jpg\" alt=\"Graph of the reciprocal function on a graphing calculator.\" width=\"975\" height=\"376\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 8<\/b><\/p>\n<\/div>\n<p id=\"fs-id1165134075625\">Based on these estimates, the function is increasing on the interval [latex]\\left(-\\infty ,-{2.449}\\right)[\/latex]<br \/>\nand [latex]\\left(2.449\\text{,}\\infty \\right)[\/latex]. Notice that, while we expect the extrema to be symmetric, the two different technologies agree only up to four decimals due to the differing approximation algorithms used by each. (The exact location of the extrema is at [latex]\\pm \\sqrt{6}[\/latex], but determining this requires calculus.)<\/p>\n<\/div>\n<\/div>\n<p><b><\/b><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1165135640967\">Graph the function [latex]f\\left(x\\right)={x}^{3}-6{x}^{2}-15x+20[\/latex] to estimate the local extrema of the function. Use these to determine the intervals on which the function is increasing and decreasing.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q48622\">Show Solution<\/span><\/p>\n<div id=\"q48622\" class=\"hidden-answer\" style=\"display: none\">\n<p>The local maximum is 28 at\u00a0<em>x\u00a0<\/em>= -1\u00a0and the local minimum is -80 at\u00a0<em>x<\/em> = 5. The function is increasing on [latex]\\left(-\\infty ,-1\\right)\\cup \\left(5,\\infty \\right)[\/latex] and decreasing on [latex]\\left(-1,5\\right)[\/latex].<\/p>\n<p><span id=\"fs-id1165134043615\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010555\/CNX_Precalc_Figure_01_03_0102.jpg\" alt=\"Graph of a polynomial with a local maximum at (-1, 28) and local minimum at (5, -80).\" width=\"487\" height=\"328\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm165724\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=165724&theme=oea&iframe_resize_id=ohm165724\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div id=\"Example_01_03_09\" class=\"example\">\n<div id=\"fs-id1165135367558\" class=\"exercise\">\n<div id=\"fs-id1165137896103\" class=\"problem textbox shaded\">\n<h3>Example 9: Finding Local Maxima and Minima from a Graph<\/h3>\n<p>For the function [latex]f[\/latex] whose graph is shown in Figure 9, find all local maxima and minima.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010555\/CNX_Precalc_Figure_01_03_0112.jpg\" alt=\"Graph of a polynomial. The line curves down to x = negative 2 and up to x = 1.\" width=\"487\" height=\"368\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 9<\/b><\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q209462\">Show Solution<\/span><\/p>\n<div id=\"q209462\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135527085\">Observe the graph of [latex]f[\/latex]. The graph attains a local maximum at [latex]x=1[\/latex] because it is the highest point in an open interval around [latex]x=1[\/latex]. The local maximum is the [latex]y[\/latex] -coordinate at [latex]x=1[\/latex], which is [latex]2[\/latex].<\/p>\n<p id=\"fs-id1165134485672\">The graph attains a local minimum at [latex]\\text{ }x=-1\\text{ }[\/latex] because it is the lowest point in an open interval around [latex]x=-1[\/latex]. The local minimum is the <em>y<\/em>-coordinate at [latex]x=-1[\/latex], which is [latex]-2[\/latex].<\/p>\n<\/div>\n<\/div>\n<p><b><\/b><\/p>\n<\/div>\n<\/div>\n<\/div>\n<section id=\"fs-id1165134544960\">\n<h1 style=\"text-align: center;\"><span style=\"text-decoration: underline;\">Analyzing the Toolkit Functions for Increasing or Decreasing Intervals<\/span><\/h1>\n<p>We will now return to our toolkit functions and discuss their graphical behavior in the table\u00a0below.<\/p>\n<table>\n<thead>\n<tr>\n<th>Function<\/th>\n<th>Increasing\/Decreasing<\/th>\n<th>Example<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>Constant Function<\/p>\n<p>[latex]f\\left(x\\right)={c}[\/latex]<\/td>\n<td>Neither increasing nor decreasing<\/td>\n<td>\u00a0<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012927\/Screen-Shot-2015-08-20-at-8.52.37-AM.png\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-12510 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012927\/Screen-Shot-2015-08-20-at-8.52.37-AM.png\" alt=\"\" width=\"143\" height=\"146\" \/><\/a><\/td>\n<\/tr>\n<tr>\n<td>Identity Function<\/p>\n<p>[latex]f\\left(x\\right)={x}[\/latex]<\/td>\n<td>\u00a0Increasing<\/td>\n<td><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012928\/Screen-Shot-2015-08-20-at-8.52.47-AM.png\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-12511 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012928\/Screen-Shot-2015-08-20-at-8.52.47-AM.png\" alt=\"\" width=\"143\" height=\"147\" \/><\/a><\/td>\n<\/tr>\n<tr>\n<td>Quadratic Function<\/p>\n<p>[latex]f\\left(x\\right)={x}^{2}[\/latex]<\/td>\n<td>Increasing on\u00a0[latex]\\left(0,\\infty\\right)[\/latex]<\/p>\n<p>Decreasing on\u00a0[latex]\\left(-\\infty,0\\right)[\/latex]<\/p>\n<p>Minimum at [latex]x=0[\/latex]<\/td>\n<td><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012928\/Screen-Shot-2015-08-20-at-8.52.54-AM.png\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-12512\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012928\/Screen-Shot-2015-08-20-at-8.52.54-AM.png\" alt=\"\" width=\"138\" height=\"143\" \/><\/a><\/td>\n<\/tr>\n<tr>\n<td>Cubic Function<\/p>\n<p>[latex]f\\left(x\\right)={x}^{3}[\/latex]<\/td>\n<td>Increasing<\/td>\n<td><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012928\/Screen-Shot-2015-08-20-at-8.53.02-AM.png\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-12513 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012928\/Screen-Shot-2015-08-20-at-8.53.02-AM.png\" alt=\"\" width=\"137\" height=\"145\" \/><\/a><\/td>\n<\/tr>\n<tr>\n<td>\u00a0Reciprocal<\/p>\n<p>[latex]f\\left(x\\right)=\\frac{1}{x}[\/latex]<\/td>\n<td>Decreasing [latex]\\left(-\\infty,0\\right)\\cup\\left(0,\\infty\\right)[\/latex]<\/td>\n<td><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012928\/Screen-Shot-2015-08-20-at-8.53.09-AM.png\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-12514 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012928\/Screen-Shot-2015-08-20-at-8.53.09-AM.png\" alt=\"\" width=\"138\" height=\"146\" \/><\/a><\/td>\n<\/tr>\n<tr>\n<td>Reciprocal Squared<\/p>\n<p>[latex]f\\left(x\\right)=\\frac{1}{{x}^{2}}[\/latex]<\/td>\n<td>Increasing on\u00a0[latex]\\left(-\\infty,0\\right)[\/latex]<\/p>\n<p>Decreasing on\u00a0[latex]\\left(0,\\infty\\right)[\/latex]<\/td>\n<td><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012928\/Screen-Shot-2015-08-20-at-8.53.16-AM.png\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-12515\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012928\/Screen-Shot-2015-08-20-at-8.53.16-AM.png\" alt=\"\" width=\"134\" height=\"145\" \/><\/a><\/td>\n<\/tr>\n<tr>\n<td>Cube Root<\/p>\n<p>[latex]f\\left(x\\right)=\\sqrt[3]{x}[\/latex]<\/p>\n<p>&nbsp;<\/td>\n<td>Increasing<\/td>\n<td>\u00a0<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012929\/Screen-Shot-2015-08-20-at-8.53.26-AM.png\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-12516\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012929\/Screen-Shot-2015-08-20-at-8.53.26-AM.png\" alt=\"Screen Shot 2015-08-20 at 8.53.26 AM\" width=\"140\" height=\"147\" \/><\/a><\/td>\n<\/tr>\n<tr>\n<td>Square Root<\/p>\n<p>[latex]f\\left(x\\right)=\\sqrt{x}[\/latex]<\/td>\n<td>Increasing on [latex]\\left(0,\\infty\\right)[\/latex]<\/td>\n<td>\u00a0<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012929\/Screen-Shot-2015-08-20-at-8.53.33-AM.png\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-12517\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012929\/Screen-Shot-2015-08-20-at-8.53.33-AM.png\" alt=\"\" width=\"138\" height=\"142\" \/><\/a><\/td>\n<\/tr>\n<tr>\n<td>Absolute Value<\/p>\n<p>[latex]f\\left(x\\right)=|x|[\/latex]<\/td>\n<td>Increasing on [latex]\\left(0,\\infty\\right)[\/latex]<\/p>\n<p>Decreasing on\u00a0[latex]\\left(-\\infty,0\\right)[\/latex]<\/td>\n<td>\u00a0<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012929\/Screen-Shot-2015-08-20-at-8.53.40-AM.png\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-12518 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012929\/Screen-Shot-2015-08-20-at-8.53.40-AM.png\" alt=\"\" width=\"135\" height=\"143\" \/><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2>Use\u00a0A Graph to Locate the Absolute Maximum and Absolute Minimum<\/h2>\n<p>There is a difference between locating the highest and lowest points on a graph in a region around an open interval (locally) and locating the highest and lowest points on the graph for the entire domain. The [latex]y\\text{-}[\/latex] coordinates (output) at the highest and lowest points are called the <strong>absolute maximum <\/strong>and<strong> absolute minimum<\/strong>, respectively.<\/p>\n<p>To locate absolute maxima and minima from a graph, we need to observe the graph to determine where the graph attains it highest and lowest points on the domain of the function. See Figure 10.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010557\/CNX_Precalc_Figure_01_03_0152.jpg\" alt=\"Graph of a segment of a parabola with an absolute minimum at (0, -2) and absolute maximum at (2, 2).\" width=\"487\" height=\"323\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 10<\/b><\/p>\n<\/div>\n<p id=\"fs-id1165137692066\">Not every function has an absolute maximum or minimum value. The toolkit function [latex]f\\left(x\\right)={x}^{3}[\/latex] is one such function.<\/p>\n<div id=\"fs-id1165135251290\" class=\"note textbox\">\n<h3 class=\"title\">A General Note: Absolute Maxima and Minima<\/h3>\n<p id=\"fs-id1165132939786\">The <strong>absolute maximum<\/strong> of [latex]f[\/latex] at [latex]x=c[\/latex] is [latex]f\\left(c\\right)[\/latex] where [latex]f\\left(c\\right)\\ge f\\left(x\\right)[\/latex] for all [latex]x[\/latex] in the domain of [latex]f[\/latex].<\/p>\n<p id=\"fs-id1165137932685\">The <strong>absolute minimum<\/strong> of [latex]f[\/latex] at [latex]x=d[\/latex] is [latex]f\\left(d\\right)[\/latex] where [latex]f\\left(d\\right)\\le f\\left(x\\right)[\/latex] for all [latex]x[\/latex] in the domain of [latex]f[\/latex].<\/p>\n<\/div>\n<div id=\"Example_01_03_10\" class=\"example\">\n<div id=\"fs-id1165134047533\" class=\"exercise\">\n<div id=\"fs-id1165134047535\" class=\"problem textbox shaded\">\n<h3>Example 10: Finding Absolute Maxima and Minima from a Graph<\/h3>\n<p>For the function [latex]f[\/latex] shown in Figure 11, find all absolute maxima and minima.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010557\/CNX_Precalc_Figure_01_03_0132.jpg\" alt=\"Graph of a polynomial.\" width=\"487\" height=\"403\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 11<\/b><\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q502428\">Show Solution<\/span><\/p>\n<div id=\"q502428\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135532371\">Observe the graph of [latex]f[\/latex]. The graph attains an absolute maximum in two locations, [latex]x=-2[\/latex] and [latex]x=2[\/latex], because at these locations, the graph attains its highest point on the domain of the function. The absolute maximum is the <em>y<\/em>-coordinate at [latex]x=-2[\/latex] and [latex]x=2[\/latex], which is [latex]16[\/latex].<\/p>\n<p id=\"fs-id1165137863670\">The graph attains an absolute minimum at [latex]x=3[\/latex], because it is the lowest point on the domain of the function\u2019s graph. The absolute minimum is the <em>y<\/em>-coordinate at [latex]x=3[\/latex], which is [latex]-10[\/latex].<\/p>\n<\/div>\n<\/div>\n<p><b><\/b><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135708033\" class=\"note precalculus media\">\n<section id=\"fs-id1165135541564\" class=\"key-equations\">\n<h2>Key Equations<\/h2>\n<table id=\"eip-id1165135358784\" summary=\"..\">\n<colgroup>\n<col \/>\n<col \/><\/colgroup>\n<tbody>\n<tr>\n<td>Average rate of change<\/td>\n<td>[latex]\\dfrac{\\Delta y}{\\Delta x}=\\dfrac{f\\left({x}_{2}\\right)-f\\left({x}_{1}\\right)}{{x}_{2}-{x}_{1}}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/section>\n<section id=\"fs-id1165135481945\" class=\"key-concepts\">\n<h2>Key Concepts<\/h2>\n<ul id=\"fs-id1165135481952\">\n<li>A rate of change relates a change in an output quantity to a change in an input quantity. The average rate of change is determined using only the beginning and ending data.<\/li>\n<li>Identifying points that mark the interval on a graph can be used to find the average rate of change.<\/li>\n<li>Comparing pairs of input and output values in a table can also be used to find the average rate of change.<\/li>\n<li>An average rate of change can also be computed by determining the function values at the endpoints of an interval described by a formula.<\/li>\n<li>The average rate of change can sometimes be determined as an expression.<\/li>\n<li>A function is increasing where its rate of change is positive and decreasing where its rate of change is negative.<\/li>\n<li>A local maximum is where a function changes from increasing to decreasing and has an output value larger (more positive or less negative) than output values at neighboring input values.<\/li>\n<li>A local minimum is where the function changes from decreasing to increasing (as the input increases) and has an output value smaller (more negative or less positive) than output values at neighboring input values.<\/li>\n<li>Minima and maxima are also called extrema.<\/li>\n<li>We can find local extrema from a graph.<\/li>\n<li>The highest and lowest points on a graph indicate the maxima and minima.<\/li>\n<\/ul>\n<div style=\"line-height: 1.5;\">\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1165133052903\" class=\"definition\">\n<dt><strong>absolute maximum<\/strong><\/dt>\n<dd id=\"fs-id1165133052908\">the greatest value of a function over an interval<\/dd>\n<\/dl>\n<dl id=\"fs-id1165133052911\" class=\"definition\">\n<dt><strong>absolute minimum<\/strong><\/dt>\n<dd id=\"fs-id1165133052916\">the lowest value of a function over an interval<\/dd>\n<\/dl>\n<dl id=\"fs-id1165133052921\" class=\"definition\">\n<dt><strong>average rate of change<\/strong><\/dt>\n<dd id=\"fs-id1165133052926\">the difference in the output values of a function found for two values of the input divided by the difference between the inputs<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135264639\" class=\"definition\">\n<dt><strong>decreasing function<\/strong><\/dt>\n<dd id=\"fs-id1165135264645\">a function is decreasing in some open interval if [latex]f\\left(b\\right)<f\\left(a\\right)[\/latex] for any two input values [latex]a[\/latex] and [latex]b[\/latex] in the given interval where [latex]b>a[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135639824\" class=\"definition\">\n<dt><strong>increasing function<\/strong><\/dt>\n<dd id=\"fs-id1165135639829\">a function is increasing in some open interval if [latex]f\\left(b\\right)>f\\left(a\\right)[\/latex] for any two input values [latex]a[\/latex] and [latex]b[\/latex] in the given interval where [latex]b>a[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135536408\" class=\"definition\">\n<dt><strong>local extrema<\/strong><\/dt>\n<dd id=\"fs-id1165135536413\">collectively, all of a function&#8217;s local maxima and minima<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135536416\" class=\"definition\">\n<dt><strong>local maximum<\/strong><\/dt>\n<dd id=\"fs-id1165135412035\">a value of the input where a function changes from increasing to decreasing as the input value increases.<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135412040\" class=\"definition\">\n<dt><strong>local minimum<\/strong><\/dt>\n<dd id=\"fs-id1165135412046\">a value of the input where a function changes from decreasing to increasing as the input value increases.<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135412050\" class=\"definition\">\n<dt><strong>rate of change<\/strong><\/dt>\n<dd id=\"fs-id1165135412054\">the change of an output quantity relative to the change of the input quantity<\/dd>\n<\/dl>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-13679\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Precalculus. <strong>Authored by<\/strong>: OpenStax College. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\">http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section><hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-13679-1\">http:\/\/www.eia.gov\/totalenergy\/data\/annual\/showtext.cfm?t=ptb0524. Accessed 3\/5\/2014. <a href=\"#return-footnote-13679-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":23588,"menu_order":3,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Precalculus\",\"author\":\"OpenStax College\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-13679","chapter","type-chapter","status-publish","hentry"],"part":10705,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/13679","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/23588"}],"version-history":[{"count":8,"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/13679\/revisions"}],"predecessor-version":[{"id":16091,"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/13679\/revisions\/16091"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/10705"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/13679\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=13679"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=13679"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=13679"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=13679"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}